Backward EDS and continuous-time sequential stochastic control in finance.

Summary We study the link between backward-looking SDEs and some stochastic optimization problems and their applications in finance. In the first part, we focus on the representation by EDSR of sequential stochastic optimization problems: impulse control and optimal switching. We introduce the notion of jump-constrained EDSR and show that it provides a representation of the solutions of Markovian impulse control problems. We then link this class of EDSRs to oblique reflection EDSRs and to the process values of optimal switching problems. In the second part we study the discretization of the EDSRs involved above. We introduce a discretization of the jump-constrained EDSRs using the penalized EDSR approximation for which we obtain convergence. We then study the discretization of obliquely reflected EDSRs. We obtain for the proposed scheme a convergence speed to the continuously reflected solution. Finally, in the third part, we study an optimal portfolio liquidation problem with risk and execution cost. We consider a financial market on which an agent must liquidate a position in a risky asset. The intervention of this agent affects the market price of this asset and leads to an execution cost modeling the liquidity risk. We characterize the value function of our problem as a minimal solution of a quasi-variational inequation in the sense of constrained viscosity.